The behavior of the core hole created in molecular x-ray photoemission experiments has provided molecular scientists with a valuable window through which to probe the electronic structure and dynamics of molecules. But the answer to one fundamental quantum question—whether the core hole is localized or delocalized—has remained elusive for diatomic molecules in which both atoms are the same element. An international team of scientists from the University of Frankfurt in Germany, Berkeley Lab, Kansas State University, and Auburn University has now resolved the issue with an appropriate twist of quantum fuzziness. By means of coincident detection of the photoelectron ejected from molecular nitrogen and the Auger electron emitted femtoseconds later, the team found that how the measurements are done determines which description—localized or delocalized—is valid.

Can One “Pin Down” Electrons?

We are used to thinking of an atom as a positively charged nucleus orbited by negatively charged electrons, but what happens to the electrons when two atoms coalesce to form a molecule. In forming a chemical bond, we know that atoms share their outer electrons so that a cloud of electrons buzzes around and between both nuclei. In effect, they belong to both at the same time, so scientists say that they are “delocalized.” But is this also true for the inner-shell electrons located closer to the nucleus and not participating significantly in bonding? Are those electrons shared by the different nuclei, or do they belong to just one nucleus, i.e., are they “localized?” Scientists have hotly disputed this fundamental question over the last 50 years, while research has yielded conflicting answers.

Now, in experiments at the ALS using low-energy x rays to remove a single electron and initiate molecular fragmentation, along with a sophisticated detector that can track every emerging particle, Schöffler et al. have answered this question for a simple molecule composed of two nitrogen atoms. In the spirit of the famous wave-particle duality in which the type of measurement determines whether one observes wave or particle-like behavior, combined with a dose of quantum weirdness in which measurement of one particle can determine the properties of another, the researchers discovered that delocalized and localized descriptions are equally valid—it all depends on how one looks!

Artist’s view of the asymmetric angular distributions obtained by coincident detection of the photoelectron emerging from a N2 molecule (blue) of the photoelectron (left) and the Auger electron emitted 7 fs later (right) for the case of a localized core hole in a N2 molecule. As the two electrons form an entangled state, the Auger electron (right) is also localized.

In order to investigate the transition between an atomic (localized) and a molecular (delocalized) description of bound electrons in N2, the researchers made use of circularly polarized photons with an energy 9 eV above the nitrogen 1s-threshold provided by ALS Beamline 11.0.2 to remove one of the innermost electrons from the nitrogen molecule. The photoelectron leaves behind a vacancy in the inner core shell N2+(1s-1), which within 7 fs is filled by an outer shell electron, resulting in the emission of a second electron (an Auger electron) carrying the excess energy. When the photoelectron and Auger electron are detected in coincidence, the Auger electron acts as a probe that in principle can determine exactly where the original hole was created.

The researchers used the Cold Target Recoil Ion Momentum Spectroscopy (COLTRIMS) technology to measure the three-dimensional momentum vectors of all four particles simultaneously (angular distribution patterns). Whether an electron is localized or delocalized is encoded in the emission pattern for the ejected electrons; however, to obtain a valid answer, the complete system must be taken into account (photoelectron, Auger electron and N2++ ionic state). Experimentally, one measures the coincident distribution of the photoelectron for a fixed direction of the Auger electron relative to the molecular axis and of the Auger electron for a fixed direction of the photoelectron, as well as the overall (non-coincident) distribution pattern for both electrons.

Ultrafast probing of a localized core hole by coincident detection of the photoelectron and Auger electron emerging from a N2 molecule. Left: Here, the photoelectron (red) is emitted from a 1s state in the left atom. About 7 fs later, the core hole decays [is filled by an electron (blue) from a higher-lying molecular p state], resulting in the emission of an Auger electron (also blue). If the core hole were delocalized, the photoelectron would come from a molecular 1s state. The two cases can be distinguished from the angular distribution patterns of the two electrons, as shown below.

In this way, the team was able for the first time to identify the existence of a Bell (entangled) state formed by the photoelectron and the Auger electron. In an entangled state, the two electrons are linked in such a way that one cannot be described without reference to the other. In the simplest cases, this means that as soon as a property of one is measured (e.g., the spin of one photon in a two-photon system with net zero spin), the corresponding property of the second is fixed as well. This feature of quantum theory, which stems from the Bell Inequality named for the late European physicist John S. Bell and provides the basis for quantum computation, allowed the team to directly address the question of localization.

Combining the entanglement feature with the symmetry of the components of the electron wave functions, it is possible for certain fixed emission directions to conclude that the innermost electron is localized, so that the second electron can then be assigned to either one of the two nuclei, which causes a right or left asymmetric emission pattern, as the case may be. For certain other fixed emission directions, it proves impossible to determine whether the first electron originated from the left or the right atom of the first electron. In this case the second electron is also delocalized, resulting in a symmetric angular distribution with respect to the molecular axis. In sum, whether you observe localized or delocalized behavior depends on how you look!